**9. Origin problem**

In the non-symmorphic space groups the symmetry elements are not constrained to pass through the origin. The selection of a reasonable origin for a coordinate system relative to non-intersecting symmetry elements is not unique. The symbol of space-group type, like Hermann-Mauguin symbol, fixes in space only relative positions of symmetry elements. Absolute positions need complete translational parts in the space-group generators. The presented technique of space group derivation based on predefined generators favours one and the only one origin for each space group, even if this group is tabulated in ITA83 relative to two origins.

Finding a transformation between two descriptions of the same space group differing only by the origin shift is arithmetically at least cumbersome. In this case, similarly like in other space-group considerations, the geometric information is very practical. A typical way of resolving the mentioned problem consists of a geometrical interpretation of the symmetry matrices in both descriptions and of deduction of the transformation from the diagram of symmetry elements in ITA83. Such analysis is impossible for a non-conventional space group description, but in every case may be carried out on dual symbols.

Since the dual symbols described in the preceding section are easy to obtain, they should be routinely derived together with the space group generation. Their role in the origin control is rather evident, but we illustrate this feature by means of an example. Let column 2 of Table 9 lists the dual symbols of the *P*42/*nnm* operations obtained from the generators presented in Table 8. The items 1, 7, 9, 15 have reduced **x**c parts. It is visible that the origin is located at the inversion point, the intersection of 2[110] and m[110]. A full symmetry of the origin is 2/m (this is the second origin choice tabulated in ITA83).

A higher symmetry of the origin can be obtain by placing it on 4�. The origin shifted from that in the first column by (1/4,-1/4,0) leads to a group description symbolized by TSG = *P*42/*nnm* (1/4,-1/4,0) (this is the first origin choice in ITA83). The result is listed in column 3 of Table 9. It can be seen that rotoinversion operations 11,12 contain the intrinsic part ½ and the inversion point according to (17) in (0,0,1/4).

Another possibility to select the origin in a high symmetry point is to put it on 4� , but exactly at the inversion point. In this situation the origin is shifted by (1/4,-1/4,1/4) in comparison with column 2. The symmetry matrices of the group description TSG = *P*42/*nnm* (1/4,-1/4,1/4) presented by dual symbols are given in the last column of Table 9.


**Table 9.** Different descriptions of the space group type *P*42/*nnm*

In order to show different group-subgroup relations, other descriptions of space groups may be desirable. Contrary to rather difficult manipulation based on coordinate triplets [28], the determination of shift vectors with the help of geometric information is simple. For this purpose the classical symbols of symmetry operations as well dual symbols are similarly useful, but the latter may be also applicable in a non-conventional space-group description. Since the multiplication of symmetry matrices is based on modulo 1 arithmetic, the origin control should involve only the generators of space groups.
